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REVIEW 2 major objections 4 minor 71 references

Superconducting Proximity Effect in Two-Dimensional Hole Gases

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A microscopic model with three interface hopping parameters yields explicit proximity-induced pairings in two-dimensional hole gases.

desk verdict A careful, self-contained microscopic derivation of the proximity effect in 2D hole gases, with real advances and honest limitations; the quantitative predictions are conditional on clean-interface assumptions that the authors themselves flag. read the letter →

arxiv 2412.04084 v2 pith:GYDUBBNW submitted 2024-12-05 cond-mat.mes-hall cond-mat.supr-con

Serafim S. Babkin , Benjamin Joecker , Karsten Flensberg , Maksym Serbyn , Jeroen Danon This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.supr-con
keywords proximityeffecttwo-dimensionalholegasgermaniumheavy-holelight-holebandsLuttinger-KohnHamiltonianinducedpairingBogoliubovFermisurfacesg-tensorrenormalization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Proximity-induced superconductivity in a two-dimensional hole gas is usually modeled by assuming the pairing is diagonal in the heavy-hole/light-hole basis. This paper argues that this assumption is too restrictive: once the interface is allowed to break rotational symmetry, the same s-wave parent pairing generates both intraband and interband pairing terms, whose magnitudes are fixed by just three real hopping amplitudes $t_x,t_y,t_z$ and the band parameters. Starting from the Luttinger-Kohn Hamiltonian and integrating out a thin superconducting film, the authors derive explicit expressions for the induced heavy-hole, light-hole, and heavy-hole-light-hole pairings, then project to the heavy-hole subspace. The effective heavy-hole Hamiltonian contains coexisting $s$-wave and $d$-wave singlet pairing, two triplet pairing terms, and superconductor-induced renormalizations of the Rashba and Zeeman couplings. A sympathetic reader cares because the result turns a phenomenological list of allowed pairings into a parameter-light, quantitative theory that can be confronted with tunneling spectra, Bogoliubov Fermi surfaces, and quantum-dot $g$-tensor measurements in germanium devices.

What carries the argument

The engine of the calculation is the Luttinger-Kohn Hamiltonian for the $J=3/2$ valence bands of the hole gas, coupled to a thin $s$-wave superconducting film by three real hopping amplitudes $t_x,t_y,t_z$ between the superconductor's $s$-orbitals and the $p_x,p_y,p_z$ orbitals of the semiconductor. The authors integrate out the superconductor in a functional integral, which replaces it by a self-energy built from the superconductor Green function, and define the dimensionless tensor $\tau_{\alpha\beta}=t_\alpha t_\beta/(|\Delta|^2+\xi_{k,s}^2-\varepsilon^2)$ that carries all quantitative information about the interface. A Schrieffer-Wolff transformation then projects the $8\times8$ Green function onto the heavy-hole $j_z=\pm3/2$ subspace, producing the effective Hamiltonian whose pairing and normal-state terms are the paper's main output.

What would settle it

Measure the tunneling density of states of a proximitized planar germanium hole gas with known confinement thickness and carrier density: the model predicts, from Eq. (45), a gap anisotropy whose extrema produce one discontinuity and one logarithmic Van Hove singularity per Rashba-split band at zero field, with the peak positions shifting and doubling when an in-plane field comparable to $0.05\Delta$ is applied. Observation of only the standard BCS square-root singularities, or a pocket pattern under in-plane field rotation that merely rotates with the field, would falsify the model's central claim about momentum-dependent pairing.

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Extended reading notes

Core claim

Integrating out the superconductor produces an explicit $8\times 8$ self-energy in the heavy-hole/light-hole space whose pairing block, Eq. (27), is fully determined by four amplitudes: $\Delta_H = -\Delta\,\tau_{+-}$, $\Delta_L = -(\Delta/3)(\tau_{+-}+2\tau_{zz})$, $\Delta_{HL,a} = \sqrt{2/3}\,\Delta\,\tau_{-z}$, and $\Delta_{HL,b} = (\Delta/\sqrt3)\,\tau_{--}$, with $\tau_{\alpha\beta} = t_\alpha t_\beta/(|\Delta|^2+\xi_{k,s}^2-\varepsilon^2)$. In the experimentally relevant case where only the heavy-hole band crosses the chemical potential, a Schrieffer-Wolff projection yields an effective heavy-hole BdG Hamiltonian whose pairing matrix, Eqs. (34)-(35), contains a direct $s$-wave term, a $d$-wave term proportional to kinetic HH-LH mixing $\zeta_F$, and two triplet terms proportional to Rashba mixing $\zeta_R$. The same projection generates anisotropic Fermi-surface deformations and additional Zeeman and Rashba terms, all controlled by the same three hopping parameters rather than by free phenomenological pairing constants.

Load-bearing premise

The derivation assumes the superconductor-hole-gas interface is clean and translationally invariant, so in-plane momentum is conserved and only one transverse mode of the superconductor couples resonantly; if the real interface is strongly disordered, the momentum-dependent pairing terms could be washed out and the quantitative link to hopping parameters would change.

Editorial extensions

If this is right

  • The induced pairing in a proximitized 2DHG is generically not diagonal in the heavy-hole/light-hole basis; interband pairings of order $\tau_{-z}$ and $\tau_{--}$ appear whenever in-plane hopping $t_\pm$ and out-of-plane hopping $t_z$ are both nonzero.
  • In the heavy-hole-only limit the proximity effect yields coexisting $s$-wave and $d$-wave singlet pairing plus $p$-wave triplet terms, so transport and spectroscopy should show angle-dependent gap anisotropy rather than a simple isotropic gap.
  • The density of states of the Rashba-split heavy-hole bands acquires logarithmic Van Hove singularities rather than BCS square-root singularities, with positions tunable by an in-plane magnetic field, and strong fields produce gapless Bogoliubov Fermi surfaces whose pocket pattern rotates non-trivially with field direction.
  • The effective $g$-tensor of a proximitized hole quantum dot is renormalized and anisotropic, with in-plane components rotated by the phase $\phi_t$ of the hopping $t_+$; measuring the Zeeman splitting as a function of field direction can therefore constrain the interface parameters.
  • All predictions depend on only three real hopping parameters in addition to known band parameters, so a small number of experiments could over-constrain the model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If real germanium interfaces are strongly disordered, the momentum-conservation assumption used here may fail; the momentum-dependent $d$-wave and triplet terms could average out, although the $s$-wave term and the parameter counting might survive in a coarse-grained theory. This is an inference, since the paper only flags disorder as a limitation.
  • The same three-parameter construction should carry over to other $p$-orbital valence-band materials, and possibly to transition-metal dichalcogenides, where orbital character is similarly entangled with spin; the paper hints at this extension without deriving it.
  • The predicted magnetic-field-orientation dependence of the Bogoliubov Fermi surfaces offers a direct test that could distinguish this heavy-hole model from earlier Rashba electron-gas treatments: a field rotation should change the number of Fermi-surface pockets instead of merely rotating them.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript develops a microscopic description of the superconducting proximity effect in a two-dimensional hole gas coupled to an s-wave superconductor. Starting from the Luttinger–Kohn Hamiltonian for the J=3/2 valence-band subspace, the authors introduce three real hopping parameters (tx, ty, tz) coupling the superconductor s-orbitals to the px, py, pz orbitals of the hole gas. They integrate out the superconductor in a path-integral formalism, obtaining a self-energy and an effective BdG Hamiltonian in the heavy-hole/light-hole basis. The central results are explicit expressions for the intraband and interband induced pairings, Eqs. (29a)-(29d), in terms of the dimensionless quantities ταβ; projection onto the heavy-hole subspace via a Schrieffer–Wolff transformation yields coexisting s-wave and d-wave singlet pairings and triplet-type terms, Eqs. (34)-(35), together with renormalized Rashba and Zeeman couplings. The paper then uses these results to compute observables: the density of states with logarithmic singularities, Bogoliubov Fermi surfaces in strong in-plane magnetic fields, and the effective g-tensor of a proximitized quantum dot.

Significance. If the central derivation is accepted, the paper makes a useful and nontrivial advance: it replaces the common assumption of pairing diagonal in the heavy-hole/light-hole basis with a quantitative relation between interface hopping parameters and the induced pairing matrix. The microscopic derivation in Appendix A is explicit and self-contained, and the reduction of the interface physics to three hopping parameters is a genuine simplification that could be constrained by tunneling and g-tensor experiments. The paper also produces concrete, falsifiable predictions, including the angular structure of the gap, magnetic-field-tunable singularities in the density of states, orientation-dependent Bogoliubov Fermi surfaces, and proximity-induced g-tensor renormalization in quantum dots. These strengths justify publication if the domain of validity of the quantitative claims is made precise.

major comments (2)
  1. [Sec. V, 'Disorder' paragraph] The clean-interface assumption is load-bearing for the quantitative core of the paper. Equations (29a)-(29d) are derived under in-plane momentum conservation (Sec. III A 3), and the angular factors in Eqs. (34)-(35) depend on the phase φt of t+ and on the relative weights of τ+− and τz+. The statement that disorder 'will not significantly change the main physical results because of the typically small semiconductor Fermi wave vector' is not a controlled derivation: it does not show how the disorder-averaged self-energy renormalizes the τ parameters or the phase φt. Since the abstract's quantitative-relationship claim and the Sec. IV predictions are based on the clean-interface expressions, a disorder-averaged calculation, or at least a quantitative estimate of the phase and weight renormalization, is needed before those predictions can be regarded as robust.
  2. [App. A, Eqs. (A7)-(A9)] The extension to multiple transverse modes preserves the form of the self-energy only under the assumption that the relative scaling between t+, t−, and tz remains unchanged as the mode index nz varies. This condition is stated but not justified. Different transverse modes of the superconducting film sample different interface wavefunctions, so the ratios t+,nz/tz,nz will generally vary with nz; in that case the induced pairings are no longer fixed by three parameters and Eq. (29) ceases to be the full quantitative statement. The paper should either justify this condition or explicitly restrict the quantitative claim to the single-resonant-mode regime, which is the case emphasized in Sec. III A 3.
minor comments (4)
  1. [Fig. 2 caption] The caption contains the text 'V4: Implemented Jeroen's idea + Dasha's suggestion+fonts', which appears to be an internal editing note and should be removed before publication.
  2. [Ref. [10]] Reference [10] lists the year as '20156'; this should be corrected to '2016'.
  3. [Sec. III A 1, Eq. (20)] The hopping Hamiltonian is projected onto the J=3/2 subspace only, discarding the J=1/2 split-off components of the pz orbital. This is likely justified by the large 3ESO energy separation, but the authors should state explicitly that the split-off channel is dropped for this reason and that any residual contribution is suppressed by Δ/3ESO rather than by the same resonant denominator used for the J=3/2 channels.
  4. [Eq. (34)] The brace labeled SP1 groups together the genuine s-wave term and a magnetic-field-induced correction that the text says is not discussed; this grouping is potentially confusing and should be clarified or relabeled.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the pairing expressions are derived by explicitly integrating out the superconductor, and the three hopping parameters are un-fitted inputs rather than quantities tuned to reproduce the predicted pairings.

full rationale

The derivation chain is self-contained. The model is defined in Sec. III A: the Luttinger-Kohn Hamiltonian for the 2DHG (Eq. 16), the BCS superconductor (Eq. 17), and three real hopping amplitudes t_x, t_y, t_z (Eq. 19). The superconductor is integrated out in App. A1, giving the self-energy Sigma = T-dagger G_SC T (Eq. A4). The induced pairings in Eqs. (29a)-(29d) are read off from the Delta-proportional block of this self-energy; they depend on the introduced tau_alpha_beta parameters but are not fitted to any target pairing. The projection onto the heavy-hole subspace is performed by an explicit Schrieffer-Wolff transformation in App. A3, yielding Eqs. (34)-(35), with the s-wave, d-wave, and triplet terms arising with computed prefactors tau_+-, tau_z+, zeta_F, and zeta_R. No equation equates an input to an output by construction: the three hoppings remain free model parameters, and Sec. IV uses illustrative, not fitted, parameter values for the DOS and g-tensor plots. The self-citations, such as [51] for the Schrieffer-Wolff transformation and [55,59] for comparison InAs and thickness effects, are not load-bearing: the transformation is explicitly derived in the appendix, and the cited works are used for context rather than to justify the central reduction. The clean-interface and single-resonant-mode assumptions are stated in Sec. III A3 and App. A, and their violation is acknowledged in Sec. V; this is a validity limitation, not a circular step. Overall, the central claim reduces neither to fitted inputs nor to a self-citation chain.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The model introduces three phenomenological interface hopping parameters as free inputs, together with the Rashba coupling and material parameters taken from literature. No new particles or fields are postulated. All derived quantities, including the induced pairings, are expressed in terms of these inputs and standard band-structure parameters.

free parameters (2)
  • t_x, t_y, t_z = not fitted
    Interface hopping parameters between s-orbitals in the superconductor and p-orbitals in the hole gas. They are treated as phenomenological inputs that parameterize the interface. The main quantitative content of the paper is expressed in terms of them.
  • Rashba coupling strength alpha_R = 10 Delta ~ 2 meV in Fig. 5
    Strength of interfacial Rashba SOC in the hole gas, taken as an input parameter and set to an experimentally realistic value for figure illustrations.
assumptions (6)
  • domain assumption Luttinger-Kohn Hamiltonian describes the valence band of Ge with the split-off J=1/2 band neglected.
    Standard model of semiconductor valence bands, justified by the ~300 meV spin-orbit energy. Invoked in Sec. III A 1.
  • domain assumption The interface is translationally invariant so that in-plane momentum is conserved during hopping.
    Used to write the hopping Hamiltonian in Eq. (18) with a single in-plane momentum k. The authors explicitly discuss disorder as a limitation in Sec. V.
  • domain assumption The superconducting film is described by BCS mean-field theory with a single s-wave order parameter Delta.
    Standard approximation for a conventional superconductor, used in Eqs. (2) and (17).
  • domain assumption Only a single resonant transverse mode n_z = n_z^(0) of the superconductor contributes to the hopping.
    Assumed in Sec. III A 3 and generalized in Appendix A. The authors argue it applies to thin films used in experiments.
  • domain assumption The spin, but not the total angular momentum, of the electron is conserved during hopping across the interface.
    Assumed in Sec. III A 3 and justified by a non-magnetic interface.
  • domain assumption The heavy-hole-light-hole splitting E_HL is the largest relevant energy scale, so that a Schrieffer-Wolff projection to leading order in 1/E_HL is valid.
    Used in Sec. III B 2 and Appendix A 3, with E_HL ~ 20 meV in experiments.

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Pith. "Pith review of Superconducting Proximity Effect in Two-Dimensional Hole Gases." pith.science (2026). https://pith.science/paper/GYDUBBNW

@misc{pith2026241204084,
  author       = {Pith},
  title        = {Pith review of: Superconducting Proximity Effect in Two-Dimensional Hole Gases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYDUBBNW}},
  note         = {Machine review of arXiv:2412.04084}
}
abstract

Technology involving hybrid superconductor-semiconductor materials is a promising avenue for engineering quantum devices for information storage, manipulation, and transmission. Proximity-induced superconducting correlations are an essential part of such devices. While the proximity effect in the conduction band of common semiconductors is well understood, its manifestation in confined hole gases, realized for instance in germanium, is an active area of research. Lower-dimensional hole-based systems, particularly in germanium, are emerging as an attractive platform for a variety of solid-state quantum devices, due to their combination of efficient spin and charge control and long coherence times. The recent experimental realization of the proximity effect in germanium thus calls for a theoretical description that is tailored to hole gases. In this work, we propose a simple model to describe proximity-induced superconductivity in two-dimensional hole gases, incorporating both the heavy-hole (HH) and light-hole (LH) bands. We start from the Luttinger-Kohn model, introduce three parameters that characterize hopping across the superconductor-semiconductor interface, and derive explicit intraband and interband effective pairing terms for the HH and LH bands. Unlike previous approaches, our theory provides a quantitative relationship between induced pairings and interface properties. Restricting our general model to an experimentally relevant case where only the HH band crosses the chemical potential, we predict the coexistence of $s$-wave and $d$-wave singlet pairings, along with triplet-type pairings, and modified Zeeman and Rashba spin-orbit couplings. Our results thus present a starting point for theoretical modeling of quantum devices based on proximitized hole gases, fueling further progress in quantum technology.

Figures

Figures reproduced from arXiv: 2412.04084 by the authors.

Figure 1
Figure 1. (a) Schematic visualization of the superconductor–semiconductor heterostructure, including the microscopic hopping [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Schematic visualization of the band structure of a typical semiconductor. The left panel depicts the conduction and valence bands for the 3D case in the absence of spin–orbit coupling. The separation between the bands at k = 0 is E0 ≈ 800 meV, which is much larger than all relevant energy scales in the problem. As a result, the conduction band is disregarded in what follows. The middle panel shows the band structure… view at source ↗
Figure 3
Figure 3. (a) Structure of the cartoons we use to represent [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Illustration of processes that induce corresponding [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: (a) Bogoliubov bands E (−) u,l as a function of two-dimensional momentum k for B = 0, as calculated from Eq. (44), illustrating the effect of the induced superconductivity on the one of the two spin–orbit-split bands. The bands feature a ϕk-dependent distance between t…
Figure 6
Figure 6. Figure 6: Angle-dependent Zeeman splittings in a proximi [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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Reference graph

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